YES
(ignored inputs)COMMENT translated from Cops 158
*** Computating Strongly Quasi-Reducible Parts ***
TRS:
   [ +(0,?y) -> ?y,
     +(?x,0) -> ?x,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
Constructors: {0,s}
Defined function symbols: {+}
Constructor subsystem:
   [ ]
Rule part & Conj Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
Rule part & Conj Part:
   [ +(0,?y) -> ?y,
     +(s(?x),?y) -> s(+(?x,?y)) ]
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)),
     +(?x,?y) -> +(?y,?x) ]
*** Ground Confluence Check by Rewriting Induction ***
Sort: {Nat}
Signature:
   [ + : Nat,Nat -> Nat,
     0 : Nat,
     s : Nat -> Nat ]
Rule Part:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjecture Part:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)),
     +(?x,?y) = +(?y,?x) ]
Precedence (by weight): {(+,3),(0,2),(s,0)}
Rule part is confluent.
R0 is ground confluent.
Check conj part consists of inductive theorems of R0.
Rules:
   [ +(?x,0) -> ?x,
     +(?x,s(?y)) -> s(+(?x,?y)) ]
Conjectures:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)),
     +(?x,?y) = +(?y,?x) ]
STEP 0
ES:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)),
     +(?x,?y) = +(?y,?x) ]
HS:
   [ ]
ES0:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)),
     +(?x,?y) = +(?y,?x) ]
HS0:
   [ ]
ES1:
   [ +(0,?y) = ?y,
     +(s(?x),?y) = s(+(?x,?y)),
     +(?x,?y) = +(?y,?x) ]
HS1:
   [ ]
Expand +(0,?y) = ?y
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2) ]
ES2:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(?x,?y) = +(?y,?x),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS2:
   [ +(0,?y) -> ?y ]
STEP 1
ES:
   [ 0 = 0,
     s(+(0,?y_2)) = s(?y_2),
     +(?x,?y) = +(?y,?x),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS:
   [ +(0,?y) -> ?y ]
ES0:
   [ 0 = 0,
     s(?y_2) = s(?y_2),
     +(?x,?y) = +(?y,?x),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS0:
   [ +(0,?y) -> ?y ]
ES1:
   [ +(?x,?y) = +(?y,?x),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS1:
   [ +(0,?y) -> ?y ]
Expand +(?x,?y) = +(?y,?x)
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2) ]
ES2:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS2:
   [ +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
STEP 2
ES:
   [ ?x_1 = +(0,?x_1),
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS:
   [ +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
ES0:
   [ ?x_1 = ?x_1,
     s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS0:
   [ +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
ES1:
   [ s(+(?x_2,?y_2)) = +(s(?y_2),?x_2),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS1:
   [ +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
Expand +(s(?y_2),?x_2) = s(+(?x_2,?y_2))
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)) ]
ES2:
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS2:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
STEP 3
ES:
   [ s(?y) = s(+(0,?y)),
     s(+(s(?y),?y_2)) = s(+(s(?y_2),?y)),
     +(s(?x),?y) = s(+(?x,?y)) ]
HS:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
ES0:
   [ s(?y) = s(?y),
     s(s(+(?y_2,?y))) = s(s(+(?y,?y_2))),
     s(+(?y,?x)) = s(+(?x,?y)) ]
HS0:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
ES1:
   [ ]
HS1:
   [ +(s(?y_2),?x_2) -> s(+(?x_2,?y_2)),
     +(?x,?y) -> +(?y,?x),
     +(0,?y) -> ?y ]
Conj part consisits of inductive theorems of R0.
examples/fromCops/cr/158.trs: Success(GCR)
(14 msec.)